Electronic structure and correlations in planar trilayer nickelate Pr4Ni3O8

The discovery of superconductivity in planar nickelates raises the question of how the electronic structure and correlations of Ni1+ compounds compare to those of the Cu2+ cuprate superconductors. Here, we present an angle-resolved photoemission spectroscopy (ARPES) study of the trilayer nickelate Pr4Ni3O8, revealing a Fermi surface resembling that of the hole-doped cuprates but with critical differences. Specifically, the main portions of the Fermi surface are extremely similar to that of the bilayer cuprates, with an additional piece that can accommodate additional hole doping. We find that the electronic correlations are about twice as strong in the nickelates and are almost k-independent, indicating that they originate from a local effect, likely the Mott interaction, whereas cuprate interactions are somewhat less local. Nevertheless, the nickelates still demonstrate the strange-metal behavior in the electron scattering rates. Understanding the similarities and differences between these two families of strongly correlated superconductors is an important challenge.

consistent with the findings from cuprates and with the previous polarized X-ray absorption spectroscopy study on Pr4Ni3O8 (22). s pol. (odd) p pol. (even) dx 2 -y 2 (odd) Allowed Forbidden dz 2 (even) Forbidden Allowed Table S1| The matrix elements of states with dx 2 -y 2 and dz 2 orbital character along the Γ-M mirror plane. be resolved from the symmetrized EDCs. Thus, in contrast to the La4Ni3O10 that displays temperature-dependent energy gaps on the dz 2 orbital band (16), the Pr4Ni3O8 hosts a gapless Fermi surface with dominant dx 2 -y 2 orbital polarization.

S3. Photon Energy Scan
To search for the extra band splitting in the DFT result, we performed a photon energy scan along the antinodal cut (X-M cut, Fig. S3a) where the band splitting is maximized. Fig. S3b shows the Fermi surface along kz and kx direction (photon energy ranging from 30 to 130eV). There are three high intensity peaks centered at kx=0, which can be clearly resolved from the spectral weight integrated along the kx direction (Fig.3c). The separation between these peaks along kz matches well with the 2*kc periodicity where kc=2 /c (c=25.5Å is the out of plane lattice constant).
However, the kF separation is very small compared to the broadening of the spectral peak, and thus, our data cannot clearly resolve the extra band splitting.

S4. Fermi Velocity Renormalization
The ratio of Fermi velocity between the DFT bands and the ARPES dispersions can also reflect the mass enhancement effect. Consider a parabolic dispersion: The Fermi velocity is the first derivative of the band dispersion at the Fermi momentum kF : As kF of the ARPES dispersions and the DFT bands are well aligned (see Fig. 3a-c in the main text), the mass enhancements with respect to the DFT band is then equal to the inverse ratio of the

S5. Spectral Function and Self-energy
ARPES is a direct measurement of the electron spectral function. which can in general be written as: where Σ′ and Σ″ are the real and imaginary parts of the electron self-energy, and k is the bare band dispersion. With the assumption of a linear bare dispersion k=vb(k-kF), a reasonable approximation when the band bottom/band top are far from the Fermi level, the spectral function can then be expressed as: For a Momentum Distribution Curve (MDC) of constant energy ω, the spectral function can be rewritten as a Lorentzian functional form: where Γ is the full width half maximum (FWHM) of the MDC. This shows that the imaginary part of the self-energy Σ″ is directly proportional to FWHM of MDC: It is worth noting that, in Fig. S4 b, the ARPES dispersion of cut 2 displays a nice fit to the linear function, even though the linear fit is constrained to the first 20 meV. Thus, the dispersion of cut 2 is an approximately linear dispersion, and we can use the method discussed above to extract the imaginary self-energy Σ″(ω) of cut 2, which is the ARPES cut in the main text for which we present the detailed analysis of Σ″(ω).
In Fig. 3E of the main text, we fit the Σ″(ω) from ARPES to the marginal Fermi liquid model described in Eqn. 3 of the main text. The extracted parameter reflecting the rising slope of Σ″(ω) is λ= 4.2±0.1, whereas similar parameters extracted from multiple dopings of Bi2212 is only at 0.5 according to our data and previous study (28). The temperature scaling factor β=7.3±0.8, and βkBT=14 meV, where kB is the Boltzmann constant, T=22 K is data-taking temperature. Thus, beyond about 15meV, the Σ″(ω) present a nearly linear energy dependence.

S6. Self-energy (Σ′ and Σ″) and mass enhancement
Here we show how the real part of the self-energy renormalizes the band dispersion by adding Σ′ to the bare band term. The spectral function in Eqn. (S3) can be rewritten in terms of the renormalization factor Z as: The pole of the spectral function is ω= k/Z, where Z=1-Σ′/ω denotes the renormalization to the bare band dispersion k. This is also the expression of the mass enhancement with respect to the bare band: The mass enhancement is thus directly proportional to the real part of the self-energy Σ′. On the other hand, the real and imaginary part of the self-energy follow the Kramers-Kronig relation: here we assume Σ″(ω′) = ω′ consistent with the marginal Fermi liquid model at zero temperature, and ωc is the cut-off energy. Although the exact solution of this integral depends on the cut-off energy (typically considered to be around the bare band bottom energy), one can tell that the real part of the self-energy Σ′ is proportional to the linear slope in Σ″, with the larger naturally corresponding to stronger band renormalization, and thus, larger mass enhancement (Eqn. S7).
Therefore, in Fig. 4 of the main text, the larger slope of Σ″ in Pr4Ni3O8 corroborates with the larger effective mass compared to other cuprate samples, even though these two properties of the manybody system--i.e., the quasiparticle scattering rate and the mass renormalization, are extracted from different aspects of the ARPES spectra (MDC widths and dispersion).

S7. DFT result of Pr4Ni3O8
The electronic structure of Pr4Ni3O8 was calculated using a conventional I4/mmm tetragonal unit cell with lattice constants a=3.9347Å, c=25.485Å, using the same method as described for La4Ni3O8 in the main paper.

S9. Mass enhancement of cuprates
The mass enhancements of the cuprates shown in the main text are consistent with previous studies in the normal state strange-metal phase (31,32,33,47,48), which showed that the normal-state antinodal mass enhancement has a similar value (47). Below the superconducting transition temperature, however, the antinodal effective mass of cuprates is enlarged to a value ~7 due to an additional strong low-energy renormalization or "kink" features (33,47); Pr4Ni3O8 does not show signs of superconductivity and displays a smooth dispersion that shows no sign of a kink anomaly.
However, we do not exclude the possibility that higher resolution measurement in future studies may reveal hidden subtle feature in the dispersion. We also note that certain cuprate compounds display higher effective masses at the quantum limit under extreme conditions of low temperature and high external magnetic field (49,50). In the main text, we only focused on the comparison to the strange metal state/normal state of the cuprate, which is the major precursor state of high TC superconductivity.

S10. Resistivity of Pr4Ni3O8
The absolute value of resistivity at room temperature is relatively large (on the scale of W cm).  Table S2. Measured resistivity at 300 K for specimens of parent Pr4Ni3O8 and La doped variants.
Data for x=0 (Sample #2 and Sample #3) are shown in Fig. S12. They of comparable magnitude with no systematic doping dependence. It is likely that the extrinsic contribution (possibly from microcracks) dominates the resistance at all temperatures and thus mask the potential trend towards metallicity on the Pr rich side of the phase diagram.

S11. Spectral linewidth and DFT band splitting
The exact linewidth at EF of Pr4Ni3O8 is ~0.12 Å -1 (FWHM of MDC in Fig. 2D). The largest band splitting from the DFT calculation is ~0.05 Å -1 (at EF), which is smaller than the spectral linewidth and potentially make it difficult to resolve. However, the spectral linewidth of Pr4Ni3O8 at the Fermi level is not large, and it is comparable to the linewidth of the cuprate samples (see Fig. 4).
In the inset of Fig. 4E, we compare the raw S¢¢ of the Pr4Ni3O8 and the cuprate samples. One can see that the impurity scattering rate (scattering rate at the Fermi level) of Pr4Ni3O8 is very close to that of the La2-xSrxCuO2 samples. Although the broadening from the impurity scattering is relatively low, the dramatic increase of the scattering rate due to the dynamic effect (the rising slope of S¢¢ in energy) certainly lead to a heavily broadened band feature and hinder the observation of two band that is close to degenerate.      photon also reveal the same electron pocket as shown in 84 eV photon data. Thus, for simplification, we used the photon energy to categorize the data in the main text. Fermi level, there is a dramatic upturn of the spectral weight and the intensity at 1eV below EF is over two orders of magnitude larger than the spectral intensity near EF. This extremely strong spectral intensity is potentially given by the Pr f states.